Mathematical Sciences: p-adic Automorphic Forms

这项研究涉及p-进对称空间的研究，模形式的算法，以及数论和代数几何中的一些计算和算法问题。在p进自同构型的一般区域中，把p进上半平面上的解析函数与测度之间的关系及其边界上的测度的已知结果推广到Drinfeld的高维p进上半空间.这些结果对由Drinfeld空间统一的代数簇的几何有一定的影响，包括某些Shimura簇和Drinfeld模簇。它们对一般线性群的p-进表示理论也有重要的结果。在模形式领域，首席调查员正在继续研究与“例外零猜想”有关的问题。最后，推广了前面关于Picard-Fuchs微分方程组和Gauss-Manin联系算法的工作。本项目的研究涉及算术几何和自同构形的一般领域。代数几何是现代数学中最古老的部分之一。在过去的十年里，它已经蓬勃发展到解决了几个世纪以来一直存在的问题。最初，它处理由最简单的方程定义的平面上的图形，即多项式。今天，这个领域不仅使用代数的方法，而且还使用分析和拓扑学的方法；相反，它在这些领域被广泛使用。此外，它在物理、理论计算机科学、密码学、编码理论和机器人学等领域都证明了自己的有用。自同构形起源于19世纪中叶的非欧几里德几何。因此，数学家和物理学家都早已认识到，许多具有根本重要性的物体本质上都是非欧几里得的。这个领域主要涉及关于整数的问题，但在几何和分析的使用中，它保留了与其历史根源的联系，从而与理论物理中的规范理论和信息论中的编码理论等不同领域的问题相联系。这项研究涉及p-进对称空间的研究，模形式的算法，以及数论和代数几何中的一些计算和算法问题。在p进自同构型的一般区域中，把p进上半平面上的解析函数与测度之间的关系及其边界上的测度的已知结果推广到Drinfeld的高维p进上半空间.这样的结果对由Drinfeld空间统一的代数簇的几何影响不大，包括某些Shimura簇和Drinfeld模簇。它们对一般线性群的p-进表示理论也有重要的结果。在模形式领域，首席调查员正在继续研究与“例外零猜想”有关的问题。最后，推广了前面关于Picard-Fuchs微分方程组和Gauss-Manin联系算法的工作。本项目的研究涉及算术几何和自同构形的一般领域。代数几何是现代数学中最古老的部分之一。在过去的十年里，它已经蓬勃发展到解决了几个世纪以来一直存在的问题。最初，它处理由最简单的方程定义的平面上的图形，即多项式。今天，这个领域不仅使用代数的方法，而且还使用分析和拓扑学的方法；相反，它在这些领域被广泛使用。此外，它在物理、理论计算机科学、密码学、编码理论和机器人学等领域都证明了自己的有用。自同构形起源于19世纪中叶的非欧几里德几何。因此，数学家和物理学家早就认识到，许多具有根本重要性的物体本质上都是非欧几里得的。这个领域主要涉及关于整数的问题，但在使用几何和分析时，它保留了与其历史根源的联系，从而与理论物理中的规范理论和信息论中的编码理论等不同领域的问题有关。

This research involves the study of p-adic symmetric spaces, the arithmetic of modular forms, and a number of computational and algorithmic questions in number theory and algebraic geometry. In the general area of p-adic automorphic forms, known results about the relationship between analytic functions and measures on the p-adic upper half plane and measures on its boundary are generalized to Drinfeld's higher dimensional p-adic upper half spaces. Such results have implications for the geometry of algebraic varieties which are uniformized by the Drinfeld spaces, including certain Shimura varieties and Drinfeld modular varieties. They also have important consequences for the p-adic representation theory of the general linear group. In the area of modular forms, the principal investigator is continuing to study questions relating to the "exceptional zero conjecture". Finally, earlier work on algorithms for computing Picard-Fuchs differential equations and the Gauss-Manin connection are extended. The research in this project lies in the general areas of arithmetic geometry and automorphic forms. Algebraic geometry is one of the oldest parts of modern mathematics. In the past ten years, it has blossomed to the point where it has solved problems that have stood for centuries. Originally, it treated figures in the plane defined by the simplest of equations, namely polynomials. Today, the field utilizes methods not only from algebra, but also from analysis and topology; conversely, it is extensively used in those fields. Moreover, it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics. Automorphic forms arose out of non-Euclidean geometry in the middle of the nineteenth century. Both mathematicians and physicists have thus long realized that many objects of fundamental importance are non-Euclidean in their basic nature. This field is principally concerned with questions about the whole numbe rs, but in its use of geometry and analysis, it retains connection to its historical roots and thus to problems in areas as diverse as gauge theory in theoretical physics and coding theory in information theory. This research involves the study of p-adic symmetric spaces, the arithmetic of modular forms, and a number of computational and algorithmic questions in number theory and algebraic geometry. In the general area of p-adic automorphic forms, known results about the relationship between analytic functions and measures on the p-adic upper half plane and measures on its boundary are generalized to Drinfeld's higher dimensional p-adic upper half spaces. Such results hate implications for the geometry of algebraic varieties which are uniformized by the Drinfeld spaces, including certain Shimura varieties and Drinfeld modular varieties. They also have important consequences for the p-adic representation theory of the general linear group. In the area of modular forms, the principal investigator is continuing to study questions relating to the "exceptional zero conjecture". Finally, earlier work on algorithms for computing Picard-Fuchs differential equations and the Gauss-Manin connection are extended. The research in this project lies in the general areas of arithmetic geometry and automorphic forms. Algebraic geometry is one of the oldest parts of modern mathematics. In the past ten years, it has blossomed to the point where it has solved problems that have stood for centuries. Originally, it treated figures in the plane defined by the simplest of equations, namely polynomials. Today, the field utilizes methods not only from algebra, but also from analysis and topology; conversely, it is extensively used in those fields. Moreover, it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics. Automorphic forms arose out of non-Euclidean geometry in the middle of the nineteenth century. B oth mathematicians and physicists have thus long realized that many objects of fundamental importance are non-Euclidean in their basic nature. This field is principally concerned with questions about the whole numbers, but in its use of geometry and analysis, it retains connection to its historical roots and thus to problems in areas as diverse as gauge theory in theoretical physics and coding theory in information theory.